The Schwinger-boson mean-field theory is applied to the quantum ferrimagnetic Heisenberg chain. There is a ferrimagnetic long-range order in the ground state. We observe two branches of the low-lying excitation and calculate the spin reduction, the gap of the antiferromagnetic branch, and the spin fluctuation at T=0 K. These results agree with the established numerical results quite well. At finite temperatures, the long-range order is destroyed because of the disappearance of the Bose condensation. The thermodynamic observables, such as the free energy, magnetic susceptibility, specific heat, and the spin correlation at T>0 K, are calculated. The T chi(uni) has a minimum at intermediate temperatures and the spin-correlation length behaves as T-1 at low temperatures. These qualitatively agree with the numerical results and the difference is small at low temperatures. [S0163-1829(99)07225-2].